Resolving closely spaced levels for Doppler mismatched double resonance
RResolving closely spaced levels for Doppler mismatched double resonance
Elijah Ogaro Nyakang’o ∗ and Kanhaiya Pandey † Department of Physics, Indian Institute of Technology Guwahati, Guwahati, Assam 781039, India (Dated: January 21, 2021)In this paper, we present experimental techniques to resolve the closely spaced hyperfine levelsof a weak transition by eliminating the residual/partial two-photon Doppler broadening and cross-over resonances in a wavelength mismatched double resonance spectroscopy. The elimination of thepartial Doppler broadening is based on velocity induced population oscillation (VIPO) and velocityselective saturation (VSS) effect followed by the subtraction of the broad background of the two-photon spectrum. Since the VIPO and VSS effect are the phenomena for near zero velocity groupatoms, the subtraction gives rise to Doppler-free peaks and the closely spaced hyperfine levels of the6P / state in Rb are well resolved. The double resonance experiment is conducted on 5S / → / strong transition (at 780 nm) and 5S / → / weak transition (at 420 nm) at room temperature. PACS numbers:
I. INTRODUCTION
Saturated absorption spectroscopy [1] is a commonlyused technique in the field of laser spectroscopy to over-come the Doppler broadening effect by canceling it inthe counter-propagation configuration of the probe andpump lasers. However, the drawback of this technique isthe formation of spurious (or cross-over) resonance peakswithin the spectrum peaks, which swamps the real res-onance peaks if the levels are closely spaced within theDoppler profile.Further, the cancellation of the Doppler effect for two-photon (or multi-photon) processes such as electromag-netically induced transparency (EIT) [2, 3] requires ap-propriate lasers propagation direction. However, thiscancellation is only possible if the wavelength of the lasersis approximately the same [3–6] otherwise suffers throughpartial Doppler broadening due to wavelengths mismatchof the transitions involved [7–10]. Recently this mismatchhas been recovered using velocity dependent light shiftfor detuned control laser and with an extra dressing laser[11, 12].In this work, we eliminate both of these problems,i.e. (i) the cross-over peaks formed within the spectrumpeaks and (ii) wavelength mismatched partial Dopplerbroadening, for double resonance at 780 nm and 420 nmof a V-type system to resolve closely spaced hyperfinelevel in Rb. The blue transition (5S / → / )at 420 nm is weak and the infrared (IR) transition(5S / → / ) at 780 nm is strong. The direct de-tection of absorption on the weak blue transition [13, 14]is a bit challenging and hence the double-resonance spec-troscopy [8, 15–18] is commonly used which again suffersthrough partial Doppler broadening. The previously dou-ble resonance spectroscopy at 420 nm and 780 nm in Rbwas mainly done in Rb due to the limitation posed by ∗ Electronic address: [email protected] † Electronic address: [email protected] the residual Doppler broadening effect [8, 15, 17, 18]. Re-solving the hyperfine levels and stabilizing the blue laserat particular transition of Rb is very important for pre-cision measurement [14, 17, 19] and laser cooling as theexpected temperature is 5 times lower in the magneto-optical trap than the routinely used IR transition, sim-ilar to the case of K [20] and Li [21]. This transition isalso useful for the coherent Rydberg excitation of Rb forquantum computation and information processing [22].The method used to overcome the above mentionedtwo problems are velocity induced population oscillationand velocity selective saturation (VSS) effects. In theatomic frame for the moving atoms, the two counter-propagating lasers with the same polarization and driv-ing the same transition, will be beating due to oppositeDoppler shift. The beating of the two lasers causes atemporal modulation of population difference betweenthe levels driven by the lasers and the phenomenon iscalled population oscillation [23–29]. Since the two beat-ing fields have same frequency, the induced populationoscillation is dependent on the velocity of the atom andhence the name velocity induced population oscillation(VIPO) [30]. The VIPO effect occurs only for a narrowrange of beat frequencies (i.e. near zero velocity range)because of the inherent population inertia i.e. the slowresponse of electric dipoles to incident fields. The rangeof beat frequencies is determined by the inverse of popu-lation relaxation times of the upper levels [29–31]. Simi-larly, VSS effect is also for near zero velocity group atomand hence the effect of partial Doppler broadening andcross-over peaks is removed for multi-photon resonance.This paper is organized in the following way. In sec-tion II, we describe the relevant energy levels with thetransitions of the various configurations and the exper-imental setup. In section III, we describe the densitymatrix formalism for the various systems considered andthe numerically simulated absorption profile of the probe.In section IV, we present the experimental results on re-solving the closely spaced hyperfine levels of the 6P / state in Rb and Rb. Finally in section V, we give theconclusion on this work. a r X i v : . [ phy s i c s . a t o m - ph ] J a n P P S F F2 F2 P u m p l a s e r , c . m P S P u m p l a s e r , c . m P P r ob e l a s e r P S P (c) g Figure 1: (Color online). Energy levels ( Rb) with hyper-fine splitting (in MHz) and the various transitions in differentconfigurations for EIT. (a) V-type open system, (b) V-typeopen system with the VIPO effect at IR transition (c) V-typeopen system with the VIPO effect at IR transition and VSSeffect at blue transition. Π g ≈
40 kHz [3] is the ground statemixing rate. P P S F F2 F2 P u m p l a s e r , c . m P S P u m p l a s e r , c . m P P r ob e l a s e r P S P (c) g Figure 2: (Color online). Energy levels ( Rb) with hyper-fine splitting (in MHz) and the various transitions in differentconfigurations for EA scheme. (a) Optical pumping system,(b) Optical pumping system with the VIPO effect at IR tran-sition (c) Optical pumping system with the VIPO effect at IRtransition and VSS effect at blue transition.
II. ENERGY LEVELS AND EXPERIMENTALSET UP
The relevant energy levels and transitions is illustratedin Fig. 1 and 2 for the V-type system and optical pump-ing system respectively. The propagation direction of theprobe and the pump (or control) lasers at 780 nm (IR)and 420 nm (blue) transitions in various configurationsis shown below the energy level scheme.The probe and the counter-propagating pump lasersat 780 nm are locked to resonance on 5S / (F = 3) ↔ / (F = 4) transition. The lifetime, τ of the state,5P / (F = 4) is 26 .
25 ns [32–34]. The absorption of theprobe is monitored as the 420 nm pump laser scans acrossthe 6P / hyperfine levels on 5S / (F = 3) ↔ / weaktransition for a V-type system or on 5S / (F = 2) ↔ / weak transition in the case of optical pumping sys-tem. The lifetime, τ of 6P / is 120 . / (F =3) ↔ / (F = 4) transition shown in Figs. 1 and 2 us-ing saturated absorption spectroscopy (SAS) set-up. Theerror signal for locking the laser is generated by frequencymodulation using the current of ECDL at 50 kHz. Therecorded experimental spectra is frequency scaled usingthe resolved peaks location of the green trace (in eachof the configurations) for the hyperfine splitting valuesgiven in reference [14].The 420 nm laser beam is generated from a commer-cially available ECDL from TOPTICA of model no. DLPRO HP with a typical linewidth of <
200 kHz andoutput power of 70 mW. A portion of the beam is fedto Fabry-Perot Interferometer for monitoring the single-mode operation of the blue laser. The beam diameter ofthe 780 nm probe and pump beams is 2 × × µ W (or peakintensity, I = 1 .
78 mW / cm ).The detailed experimental set up is shown in Fig. 3.In order to extract the narrow linewidth, the probe laserbeam is divided into two beams with same polarizationand power and propagated in the Rb cell with a spatialseparation of about 1 cm. The blue beam is also dividedinto two beams with the same polarization as the IRbeams and co-propagates with the two probes as shownin experimental set-up of Fig. 3. The IR pump beamwhich counter-propagates with one of the probe beam,has the same polarization as the probe beam since havingthe same polarization is key for the interference/beatingof the two fields. The interference/beating of the fieldsrequires the polarization of the two fields to be identicaland this aspect has been verified experimentally by ro-tating the polarization of one of the fields. When the po-larization of the two fields are orthogonal, the VIPO dipdisappear. There is a retro-mirror for reflecting the bluebeam (which is overlapping with the IR pump beam) togenerate counter-propagating blue beams in the cell whenshutter 2 is open. It is very important to keep the anglebetween the beams as minimum as possible (i.e. nearzero angle) and also use a magnetic shield to minimizebroadening of the spectrum.There are three shutters which are used to generatevarious conditions and configurations in the experiment.The configuration represented by Fig. 1a or 2a is gen-erated with all the shutters closed. The configurationrepresented by Fig. 1b or 2b is generated with shutter 1open and shutter 2 closed. The configuration representedby Fig. 1c or 2c is generated with shutter 1 and shutter2 open. Opening the shutter 3 removes the broad back-ground of the transparency and EA peaks. The broadbackground is removed by the subtraction of the absorp-tion/transparency spectra of the two probes using two Figure 3: (Color online). The experimental setup for for re-solving the closely spaced hyperfine levels of the 6P / statein Rb atom using the VIPO and VSS effects. identical IR photo-detectors (PD1 and PD2) in the dif-ferential transimpedance amplifier. III. THEORETICAL MODEL
We have conducted the experiments in the six config-urations shown in Fig. 1 and 2, three of them are for theV-type open system (Fig. 1a, 1b and 1c) and the otherthree are for optical pumping system (Fig. 2a, 2b and2c). The V-type open system is further sub-categorizedinto: (i) V-type open system shown in 1a, (ii) VIPOat IR transition for V-type open system shown in Fig.1b and (iii) VIPO at IR and VSS at blue transition forV-type open system shown in Fig. 1c. Similarly, the op-tical pumping system is sub-categorized into: (i) Opticalpumping system shown in Fig. 2a, (ii) VIPO at IR tran-sition for optical pumping system shown in Fig. 2b and(iii) VIPO at IR and VSS at blue transition for opticalpumping system shown in Fig. 2c. We discuss the theoryfor all these configurations one by one.
A. Transparency for V-type open system
1. V-type open system
This corresponds to the energy level and the configu-ration shown in Fig. 1a and is achieved by closing all theshutters of the experimental setup in Fig. 3. This sys-tem is very well known and has been extensively studied[8, 36]. This V-type of system is open as the populationfrom 6P / decays to the other ground state hyperfinelevel, 5S / (F = 2) and can not be recycled. In the pres-ence of the blue pump laser, c2, there is transparencyof the IR probe laser due to two effects, one is coherenceeffect i.e. EIT in a V-type atomic system [37, 38] and theother is optical pumping effect [39–41]. The Hamiltonianof the system, the equations of motion and the analyticalexpression for the absorption of the probe, ρ , are givenin Eq. A1, A2 and A3 respectively.The mixing rate, Π g , for the hyperfine ground states(appearing in the equations of motion) is due to thermalcollisions and the time of fight of atoms across the laserbeam [3, 42]. The contribution due to time of flight isdefined as d/ ˜ v where, ˜ v is the thermal velocity of theatoms in the atomic medium and d is the diameter ofthe laser beam. The numerically simulated absorptionspectrum of the IR probe laser locked to resonance on5S / (F = 3) ↔ / (F = 4) cycling transition vs de-tuning of the blue pump laser is plotted in Fig. 4 (see theblue trace). The Lorentzian fitting to this curve gives alinewidth of 16 MHz, while the linewidth is 11 MHz if thepump laser wavelength is taken to be 780 nm instead of420 nm (see the table I). This broadening by 1.5 times isdue to residual or partial Doppler broadening caused bywavelength mismatch between the probe and the pumplaser.
2. VIPO at IR transition for V-type open system
This configuration corresponds to the energy schemegiven in Fig. 1b and the experimental set-up when shut-ter 1 is open and shutter 2 is closed. This is theoreti-cally modeled by considering the Hamiltonian H underelectric-dipole and rotating-wave approximation and inthe interaction picture as follows, H = (cid:126) (cid:8) (Ω c1 + Ω p e iδ t ) | (cid:105)(cid:104) | + Ω c2 | (cid:105)(cid:104) |− ∆ c1 | (cid:105)(cid:104) | − ∆ c2 | (cid:105)(cid:104) | + h.c. (cid:9) (1)where, 5S / (F = 3) = | (cid:105) , 5P / (F = 4) = | (cid:105) ,6P / (F = 2) = | (cid:105) , δ = ( ω p − k v ) − ( ω c1 + k v ) = − k v is the frequency difference between IR probe and pumpbeams in the atomic frame (since ω p = ω c1 ), k = 2 π/λ is the wave-vector of the IR laser and λ is the wave-length, v is the velocity of the atom in the direction ofthe probe, ∆ c1 = ω c1 − ( ω − ω ) + k v is the detuningof the IR control laser, ∆ c2 = ω c2 − ( ω − ω ) − k v isthe detuning of the blue laser, k = 2 π/λ is the wave-vector of the blue laser and λ is the wavelength. TheRabi frequency for the fields is Ω L = − d ij E L / (cid:126) where,d ij = (cid:104) i | ˆd | j (cid:105) is the dipole matrix element, ˆd is the atomicdipole operator and subscript L = p , c1 , c2 represent thefields (i.e. p is the probe and c1 is the pump of the 780 nmlaser and c2 is the pump of the 420 nm laser).The atom-field interaction is described by writing theLiouville-von Neumann equation for the density matrix,˙ ρ = − i (cid:126) [H , ρ ] − { Γ , ρ } (2)where, ρ is the atomic density operator, Γ is the relax-ation operator defined as (cid:104) i | Γ | j (cid:105) = γ i δ ij ( δ ij = 1 if i = j and 0 if i (cid:54) = j ) and γ i is the decay rate of state | i (cid:105) . Thetemporal behavior of the element of density matrix gov-erned by Eq. 2 is velocity dependent due to the Dopplereffect and oscillates at the harmonics of the beat fre-quency δ = − k v . The oscillation is caused by thebeating of the two fields addressing the same transition5S / (F = 3) ↔ / (F = 4) in Fig. 1a. The equationsof motion of the density matrix elements is given in Eq.A4 and is obtained using Eq. 1 and 2. The harmonicallyoscillating density matrix elements at beat frequency canbe written in the Floquet expansion [43–45] in the follow-ing form ρ ij ( t ) = ∞ (cid:88) n = −∞ ρ ( n ) ij ( t ) e inδ t (3)where, ρ ( n ) ij ( t ) are n th harmonic amplitudes of the den-sity matrix elements. The imaginary part of the zerothharmonic, ρ (0)12 corresponds to the IR pump absorption,while the imaginary part of the first harmonic, ρ (+1)12 isfor IR probe absorption in first order and all the othersare for wave-mixing [46]. In the steady state condition( ˙ ρ ( n ) ij = 0 for all n, i and j ), the absorption of the probelaser ( ρ (+1)12 ) is obtained by substituting the truncatedseries of the Floquet expansion given in Eq. 3 up to first-order into Eq. A4. The coefficients of the same power in nδ are then compared which yields a set of steady stateequations of motion in the Floquet expansion. The ρ (+1)12 element of the density matrix is expressed as follows, ρ (+1)12 = I (cid:122) (cid:125)(cid:124) (cid:123) i Ω p γ + iδ ) ( ρ (0)11 − ρ (0)22 ) + i Ω c γ + iδ ) ( ρ (+1)11 − ρ (+1)22 ) (cid:124) (cid:123)(cid:122) (cid:125) II − i Ω c γ + iδ ) ρ (+1)32 (cid:124) (cid:123)(cid:122) (cid:125) III (4)where, γ = i ∆ p + γ dec , ∆ p = ∆ c = 0, γ decij = (Γ i +Γ j )and Γ i is the decay rate of the i th level. The quantity( ρ (0)11 − ρ (0)22 ) in term I is the population inversion created -100 -50 0 50 100 Detuning of 420 nm laser (MHz) P r ob e a b s o r p t i on -4 P r ob e a b s o r p t i on -4 Fig. 6.1aLorentzian fitFig. 6.1bFig. 6.1cGaussin fit -10 0 1000.51
Normalized dips
Figure 4: (Color online). Numerically calculated thermal av-eraged probe absorption vs detuning of 420 nm pump laserfor V-type open system with various configurations shown inFig. 1 (where Ω p = √ . ). The blue curve correspondsto Fig. 1a with Ω c = √ . . The red curve marked by cir-cles corresponds to Fig. 1b with Ω c = Γ and Ω c = √ . .The green curve marked with dots corresponds to Fig. 1cwith Ω c = √ . , Ω c = √ . , Γ = 2 π × .
065 MHz andΓ = 2 π × .
32 MHz. The vertical axis of the blue trace ison the left and the red trace marked by circles and the greentrace marked with dots are on the right. by the pump lasers at IR and blue transition. The quan-tity ( ρ (+1)11 − ρ (+1)22 ) in term II is the population oscillationdifference and its contribution is significant for the veloc-ity group atoms in the range of | v | ∼ Γ /k and formsa dip inside the transparency window. The density ma-trix element ρ (+1)32 in term III is the coherence oscillationwhich further modifies the lineshape of the dip inside thetransparency window. The role of individual terms forthe probe absorption is shown in Fig. A.2.The absorption of the probe laser is obtained by ther-mal averaging of Eq. 4 at room temperature as follows, √ π ˜ v (cid:82) ρ (+1)12 e − ( v v ) d v , where, ˜ v = (cid:112) k B T /m , m (= 85a.m.u) is the atomic mass and T (= 300 K) is the tem-perature. The lineshape of the probe absorption afterthermal averaging is shown in Fig. 4 (see the red tracemarked by circles). The linewidth of the dip inside thetransparency window is around 7 MHz which is less thanthe linewidth for a V system if the pump laser had wave-length at 780 nm instead of 420 nm. The linewidth ofthe dip is determined by fitting with a Gaussian line-profile (which fits better than a Lorentzian line-profile).The FWHM of a Gaussian fit (i.e. A e − ( x − x ) / (2 σ ) ), is2 √ σ where A, x and σ are the fitting parametersand x is the frequency detuning.
3. VIPO at IR and VSS at blue transition for V-type opensystem
The energy scheme for this configuration is given inFig. 1c where the probe and IR pump are similarly lockedto resonance on 5S / (F = 3) ↔ / (F = 4) cycling Table I: Comparison of numerically calculated linewidth forvarious configurationSystem and configuration Linewidth(MHz)Configuration as shown in Fig. 1a 16Configuration as shown in Fig. 1a but consideringpump wavelength 780 nm instead of 420 nm 11Configuration as shown in Fig. 1b 7 ± ± ± ± ± transition. The blue pump scans across the hyperfinelevels of the 6P / at the weak transition 5S / (F = 3) ↔ / and is retro-reflected by mirror M to generate thetwo counter-propagating beams inside the Rb vapor cell.The VIPO on 5S / (F = 3) ↔ / (F = 4) transitionwill induce a dip on the transparency peak as previouslyexplained in the Sec. III A 2. This dip is further enhancedby the VSS effect of the two counter-propagating bluepump laser beams.The VSS effect can be understood in the following sim-ple way. We consider population dynamics between thetwo states, | (cid:105) (5S / , F=2) and | (cid:105) (6P / ) due to twocounter-propagating blue pump laser beams only in theabsence of the IR laser. For simplicity, consider threevelocity group of atoms, + v , 0 and − v . For detuned caseof the blue pump laser, (∆ c ) both the non-zero velocitygroup of atom ± v = ∆ c /k will be in resonance with ei-ther of the two counter-propagating blue pump laser andhence the number of atoms in the excited state will betwice. For zero detuning case, the near-zero ( < Γ /k )velocity group of atom will be in resonance with both theblue pump laser and hence intensity seen by this groupof atoms will be twice. However, the excited state pop-ulation will be less than twice due to saturation effect,thus inducing a dip on the absorption spectrum of theprobe beam with the scan of the blue pump laser. Thelinewidth of this dip is in the range of Γ . This quali-tative picture is also presented in [47]. Mathematically,the population transfer due to blue pump lasers will begiven by the following equation [30]. ρ = 12 (cid:40) S ∆ c − k v + S ∆ c + k v ∆ c − k v + S ∆ C + k v (cid:41) (5)with, S ∆ c + k v = S c + k v ) Γ , S ∆ c − k v = S c − k v ) Γ where S (= 2Ω c / Γ ) is the saturation intensity of theblue transition for the stationary atoms. In the presenceof the IR pump laser i.e. when shutter 1 and 2 are open,the VSS effect will induce a dip on both the transparencyspectra of both the IR pump and the probe.The detailed formalism for the VIPO at IR and VSSat blue transitions is as follows. For the given velocity v there is a beating for the two-counter-propagating bluepump lasers in the atomic frame with a beat frequency( δ = − k v ). The Hamiltonian H of a V-type systemshown in Fig. 1c under electric-dipole and rotating-waveapproximation and in the interaction picture is as follows,H = (cid:126) (cid:8) (Ω c1 + Ω p e iδ t ) | (cid:105)(cid:104) | + (Ω c2 + Ω c2 e iδ t ) | (cid:105)(cid:104) |− ∆ c1 | (cid:105)(cid:104) | − ∆ c2 | (cid:105)(cid:104) | + h.c. (cid:9) (6)The equations of motion of the density matrix elements isgiven in Eq. A5 which is obtained using Eq. 2 and 6. Thecoefficients of the harmonically oscillating density matrixelements have two different time dependencies, which isalso the case for the Hamiltonian in Eq. 6. The Floquetexpansion for the density matrix elements in such a caseis modified and written as follows, ρ ij ( t ) = ∞ (cid:88) m = −∞ (cid:16) ∞ (cid:88) n = −∞ ρ ( n,m ) ij ( t ) e i ( nδ + mδ ) t (cid:17) (7)where, n is the n th harmonic component due the beatingof the IR laser beams and m is the m th harmonic com-ponent due the beating of the blue pump laser beams.The imaginary part of ρ (0 , corresponds to the IR pumpabsorption, while the imaginary part of ρ (+1 , is for IRprobe absorption and all the others are for wave-mixing.In the steady state condition (i.e. ˙ ρ ( n,m ) ij = 0 for all n , m , i and j ), ρ (+1 , is obtained by substituting the trun-cated series of the Floquet expansion given in Eq. 7 upto first-order into Eq. A5. The coefficients of the samepower in ( nδ , mδ ) are similarly compared and yields aset of steady state equations of motion in the Floquetexpansion. The ρ (+1 , element of the density matrix isexpressed as follows, ρ (+1 , = (cid:110) I (cid:122) (cid:125)(cid:124) (cid:123) i Ω p ( ρ (0 , − ρ (0 , )2( γ + iδ ) + i Ω c ( ρ (+1 , − ρ (+1 , )2( γ + iδ ) (cid:124) (cid:123)(cid:122) (cid:125) II − i Ω c ( ρ (+1 , + ρ (+1 , − )2( γ + iδ ) (cid:124) (cid:123)(cid:122) (cid:125) III (cid:111) (8)In Eq. 8, the quantity ( ρ (0 , − ρ (0 , ) in term I isthe population inversion induced by the IR pump andthe saturation of the counter-propagating blue pumpsand ( ρ (+1 , − ρ (+1 , ) in term II is the population os-cillation induced by the beating of the IR probe andpump laser beams and the saturation of the counter-propagating blue pumps. The density matrix elements, ρ (+1 , and ρ (+1 , − in term III, are the oscillating co-herence terms due to the beating of the fields on IR andblue transitions. The thermal averaged probe absorption, √ π ˜ v (cid:82) ρ (+1 , e − ( v v ) d v is calculated numerically and isplotted in Fig. 4 (see the green trace marked with dots).The linewidth of the induced dip is around 6 MHz. B. Enhanced absorption for optical pumpingsystem
1. Optical pumping system
This system corresponds to the energy level and theconfiguration shown in Fig. 2a and is achieved whenall the shutters in the experimental setup of Fig. 3 areclosed. Again, the probe laser is locked to resonanceon 5S / (F = 3) ↔ / (F = 4) transition. The ab-sorption of the probe is monitored as the co-propagatingblue pump laser scans across the 6P / hyperfine lev-els on 5S / (F = 2) → / transition instead of5S / (F = 3) → / transition. The absorption ofthe probe is increased by optical pumping of populationto the upper ground hyperfine level 5S / (F = 3) [39–41]via 5S / (F = 2) → / (F = 1 , ,
3) excitation andvarious decay channels (i.e. direct, 6P / (F = 2 , → / (F = 3) and indirect decay channels [48] such as6P / (F = 1) → / → / → / (F = 3)).Therefore, optical pumping [49, 50] gives rise to enhancedabsorption (EA) Doppler-free peaks of the 6P / hyper-fine levels. The numerically simulated absorption spec-trum considering only one hyperfine level is plotted inFig. 5 (see the blue trace). Note that the Hamiltonianof the system, equations of motion and the analytical ex-pression for the absorption of the probe, ρ , are givenin Eq. B1, B2 and B3 respectively. The Lorentzian fit-ting to this curve gives a linewidth of 17 MHz, whileit is 11 MHz if we consider the pump laser wavelengthto be 780 nm instead of 420 nm (see the table I). Thisbroadening by 1.5 times is again due to residual or par-tial Doppler broadening caused by wavelength mismatchbetween the probe and the pump laser.
2. VIPO at IR transition for optical pumping system
This corresponds to the energy level and the config-uration shown in Fig. 2b and is achieved when shutter2 is closed in the experimental setup of Fig. 3. This istheoretically modeled by considering the Hamiltonian Hof the optical pumping system shown in Fig. 2b underelectric-dipole and rotating-wave approximation and in the interaction picture as follows,H = (cid:126) (cid:8) (Ω c1 + Ω p e iδ t ) | (cid:105)(cid:104) | + Ω c2 | (cid:105)(cid:104) |− ∆ c1 | (cid:105)(cid:104) | − ∆ c2 | (cid:105)(cid:104) | + h.c. (cid:9) (9)where, 5S / (F = 3) = | (cid:105) , 5P / (F = 4) = | (cid:105) ,6P / (F = 1) = | (cid:105) and 5S / (F = 2) = | (cid:105) . The Hamil-tonian is time dependent and the equations of motionof the density matrix elements is given in Eq. B4. Theequations of motion are solved in steady state after theFloquet expansion given in Eq. 3 and the imaginary partof the density matrix element ρ (+1)12 gives the absorptionof the probe as follows, ρ (+1)12 = I (cid:122) (cid:125)(cid:124) (cid:123) i Ω p γ + iδ ) ( ρ (0)11 − ρ (0)22 ) + i Ω c γ + iδ ) ( ρ (+1)11 − ρ (+1)22 ) (cid:124) (cid:123)(cid:122) (cid:125) II (10)The Eq. 10 is similar to the Eq. 4 except the coherenceterm. The first term, I in Eq. 10 is due to populationinversion created by the pump laser at IR and blue tran-sition and gives only the EA line-shape. The second termis due to VIPO at IR transition and gives a dip inside theEA spectrum as shown Fig. 5 (see the red trace markedby circles). The linewidth of the dip is 9 MHz usingGaussian line profile fit. The contribution of each of theterms I and II is given in Fig. B.2.
3. VIPO at IR and VSS at blue transition for opticalpumping system
The energy levels for this configuration is given inFig. 2c. The probe and the IR pump lasers are againlocked to resonance on 5S / (F = 3) ↔ / (F = 4)cycling transition. The blue pump laser is scanningacross the hyperfine levels of 6P / at the weak tran-sition, 5S / (F = 2) ↔ / and is retro-reflected togenerate the two counter-propagating beams inside theRb vapor cell.The Hamiltonian H of the optical pumping systemshown in Fig. 2c under electric-dipole and rotating-waveapproximation and in the interaction picture is given asfollows,H = (cid:126) (cid:8) (Ω c1 + Ω p e iδ t ) | (cid:105)(cid:104) | + (Ω c2 + Ω c2 e iδ t ) | (cid:105)(cid:104) |− ∆ c1 | (cid:105)(cid:104) | − ∆ c2 | (cid:105)(cid:104) | + h.c. (cid:9) (11)The probe absorption is similarly obtained in the steadystate condition using the equations of motion given inEq. B5 and the Floquet expansion given in Eq. 7. Theimaginary part of the density matrix element ρ (+1 , in -100 -50 0 50 100 Detuning of 420 nm laser (MHz) P r ob e a b s o r p t i on -4 P r ob e a b s o r p t i on -4 Fig. 6.5aLorentzian fitFig. 6.5bFig. 6.5cGaussin fit -10 0 1000.51
Normalized dips
Figure 5: (Color online). Numerically calculated thermal av-eraged probe absorption vs detuning of 420 nm pump laserfor optical system with various configurations shown in Fig.2 (where Ω p = √ . ). The blue curve corresponds toFig. 2a with Ω c = √ . . The red curve marked by circlescorresponds to Fig. 2b with Ω c = Γ and Ω c = √ . .The green curve marked with dots corresponds to Fig. 2cwith Ω c = √ . , Ω c = √ . , Γ = 2 π × .
065 MHz andΓ = 2 π × .
32 MHz. The vertical axis of the blue trace is onleft and the red trace marked by circles and the green tracemarked with dots are on the right. the Floquet expansion gives the probe absorption and isexpressed as follows, ρ (+1 , = I (cid:122) (cid:125)(cid:124) (cid:123) i Ω p ( ρ (0 , − ρ (0 , )2( γ + iδ ) + i Ω c ( ρ (+1 , − ρ (+1 , )2( γ + iδ ) (cid:124) (cid:123)(cid:122) (cid:125) II (12)In Eq. 12, the quantity ( ρ (0 , − ρ (0 , ) in term I is thepopulation inversion induced by the 780 nm and 420 nmpump lasers. The quantity ( ρ (+1 , − ρ (+1 , ) in term II isthe population oscillation induced by the beating of the780 nm laser beams and saturation effect induced by thecounter-propagating 420 nm pump beams. The thermalaveraged absorption in this configuration is shown in Fig.5 (see the green trace marked with dots). The linewidthof the induced dip on the EA peak is about 6 MHz. IV. EXPERIMENTAL RESULTSA. Resolving P / hyperfine levels in Rb
1. V-type open system
The transparency spectrum of the energy scheme inFig. 1a is shown by the red dashed trace of Fig. 6.This spectrum is obtained when all the three shuttersin the experimental set-up of Fig. 3 are closed. Thethree peaks of the 6P / (F = 2 , ,
4) hyperfine levels aremerged forming a broad transparency spectrum due to -500 -400 -300 -200 -100 0 100 200 300 400 500
Frequency (MHz) of 420 nm laser P r ob e a b s o r p t i on ( a r b . un i t s ) Fig. 1aFig. 1bFig. 1b - Fig. 1a -500 -400 -300 -200 -100 0 100 200 300 400 500
Frequency (MHz) of 420 nm laser P r ob e a b s o r p t i on ( a r b . un i t s ) Fig. 1aFig. 1cFig. 1c - Fig. 1a
Figure 6: (Color online). The transparency spectrum of the6P / hyperfine levels in Rb under various configurationsshown in Fig. 1. The red dashed trace is for the V-type opensystem (Fig. 1a), the blue trace marked with dots in Fig. 6a isfor the V-type open system with VIPO effect at IR transition(Fig. 1b) while the blue trace marked with dots in Fig. 6bis for V-type open system with VIPO effect at IR and VSSeffect at blue transition (Fig. 1c). The green trace is the finalresult after removing the broad transparency background andit is magnified 3 times for visibility purpose. the residual Doppler broadening effect. When shutter 1is open, dips corresponding to three hyperfine levels areinduced inside the broad transparency peaks caused byVIPO at IR transition (see the blue trace marked withdots in Fig. 6a). However, the dips appear very smalldue to the broad transparency background. The effectis removed when shutter 3 is open to subtract the broadtransparency profile and the spectrum of the resolvedhyperfine levels is shown by the green trace of Fig. 6a.The linewidth of the resolved peaks are as follows: F = 4is 13.3 MHz, F = 3 is 14.1 MHz and F = 2 is 12.1 MHz.The power of the pump beams labeled c1, c2 and c3used for optimal signal-to-noise ratio of the spectrum are276.2 µ W (or peak intensity I=11.7 mW / cm ), 5.02 mW(or peak intensity I=106.5 mW / cm ) and 3.64 mW (orpeak intensity I=77.2 mW / cm ) respectively.Further line narrowing of the resolved peaks is achievedusing the configuration shown in Fig. 1c (i.e. VIPO atIR and VSS at blue transition). The energy configura-tion scheme in Fig. 1c (i.e. VIPO at IR and VSS atblue transition) is implemented in the experimental set-up given in Fig. 3, when shutter 1 and shutter 2 areopen. Lower power of IR pump beam is used in this -500 -400 -300 -200 -100 0 100 200 300 400 500 Frequency (MHz) of 420 nm laser P r ob e a b s o r p t i on ( a r b . un i t s ) Fig. 2aFig. 2bFig. 2b-Fig. 2a -500 -400 -300 -200 -100 0 100 200 300 400 500
Frequency (MHz) of 420 nm laser P r ob e a b s o r p t i on ( a r b . un i t s ) Fig. 2aFig. 2cFig. 2c - Fig. 2a -100 -75 -50 -25 0 25 500.40.50.60.70.8 321
Figure 7: (Color online). The EA spectrum of the 6P / hyperfine levels in Rb under various configurations shownin Fig. 2. The red dashed trace is for the optical pumpingsystem (Fig. 2a), the blue trace marked with dots in Fig.7a is for the optical pumping system with VIPO effect at IRtransition (as shown in Fig. 2b) while the blue trace markedwith dots in Fig. 7b is for the optical pumping system withVIPO effect at IR and VSS effect at blue transition (Fig.2c). The green trace is the final result after removing broadabsorption background and it is magnified 3 time for visibilitypurpose. configuration since the induced dips by VIPO at IR areenhanced by VSS effect at blue transition. The trans-parency spectrum of this configuration is shown by theblue trace marked with dots in Fig. 6b. The broad trans-parency background is removed when shutter 3 is openand the well resolved peaks of the 6P / (F = 2 , ,
4) hy-perfine levels is shown by the green trace of Fig. 6b. Thelinewidth of the resolved peaks are as follows: F = 4 is10.8 MHz, F = 3 is 9.1 MHz and F = 2 is 11.4 MHz.The power of the pump beams labeled c1, c2 and c3used for optimal signal-to-noise ratio of the spectrum are176.4 µ W (or peak intensity I=7.5 mW / cm ), 6.01 mW(or peak intensity I=127.5 mW / cm ) and 8.62 mW (orpeak intensity I=182.9 mW / cm ) respectively.In the final result of the resolved peaks (see the greentrace of Fig. 6b), there are small peaks between themain peaks of F = 3 and F = 4 and between F = 2 andF = 3. These are not cross-over peaks (or real peaks),but the residue due to incomplete removal of the broadtransparency background in the overlapped regions. Theeffect also occur for the optical pumping system when thebroad absorption background is removed (see the green trace of Fig. 7b the small peak between F = 2 and F =3).
2. Optical pumping system
The EA spectrum of the optical pumping system isshown by the red dashed trace in Fig. 7. This spectrumis obtained when all the three shutters in the experimen-tal set-up of Fig. 3 are closed. The absorption peakscorresponding to the 6P / (F = 1 , ,
3) hyperfine levelsare completely merged. The levels 6P / (F = 2 ,
3) aredetected by the probe via both the direct decay and in-direct decay channels [48] while level 6P / (F = 1) isdetected via the indirect decay channels to 5S / (F = 3)only. When shutter 1 is open, dips corresponding to thehyperfine levels are induced inside the broad EA peaksdue to VIPO at IR transition (see the blue trace markedwith dots in Fig. 7a). The dips appear small due tobroad EA background caused by the residual Dopplerbroadening effect. The broad EA background is removedwhen shutter 3 is open and the dips corresponding tothe hyperfine levels 6P / (F = 1 ,
2) are still not re-solved while the 6P / (F = 3) peak is resolved (see thegreen trace of Fig. 7a). The linewidth of the resolvedpeak is F = 3 is 13.9 MHz. The power of the pumpbeams labeled c1, c2 and c3 used for optimal signal-to-noise ratio of the spectrum are 806.2 µ W (or peakintensity I=34.2 mW / cm ), 5.01 mW (or peak inten-sity I=106.3 mW / cm ) and 1.84 mW (or peak intensityI=39.0 mW / cm ) respectively.The peaks corresponding to the 6P / (F = 1 , ,
3) hy-perfine levels, can be completely resolved using the con-figuration shown in Fig. 2c i.e. VIPO at IR and VSSat blue transition. This configuration is implementedwhen shutter 1 and shutter 2 are open in the experimen-tal set-up of Fig. 3. The broad EA spectrum is removedwhen shutter 3 is open and the green trace of Fig. 7bshows well resolved peaks of the 6P / (F = 1 , ,
3) hy-perfine levels. Note, the frequency scaling of the spectrain Fig. 7 is assigned using the peak locations of F = 2 andF = 3 after the complete resolution of all the three peaksof 6P / (F = 1 , ,
3) hyperfine levels. The linewidth ofthe resolved peaks are as follows: F = 3 is 9.8 MHz,F = 2 is 10.1 MHz and F = 1 is 7.2 MHz. The powerof the pump beams labeled c1, c2 and c3 used for opti-mal signal-to-noise ratio of the spectrum are 276.3 µ W(or peak intensity I=11.7 mW / cm ), 5.02 mW (or peakintensity I=106.7 mW / cm ) and 15.19 mW (or peak in-tensity I=322.3 mW / cm ) respectively.Besides the main peaks due to near zero-velocity groupatoms in Fig. 7a, the extra peaks (or cross-over peaks)formed outside the main spectrum are caused by atomsmoving with velocities of 94 ms − and 143 ms − respec-tively. Atoms moving with velocities of 94 ms − and143ms − along the propagation direction of the IR probe,will see the probe laser to be on resonance with the5S / (F = 3) → / (F = 3) and 5S / (F = 3) → -500 -400 -300 -200 -100 0 100 200 300 400 500 Frequency (MHz) of 420 nm laser P r ob e a b s o r p t i on ( a r b . un i t s ) Fig. 1aFig. 1cFig. 1c - Fig. 1a
Figure 8: (Color online). The transparency spectrum of the6P / hyperfine levels in Rb recorded for similar configura-tions of Rb shown in Fig. 1. The red dashed trace is for theV-type open system (Fig. 1a), the blue trace marked withdots is for V-type open system with VIPO effect at IR andVSS effect at blue transition (Fig. 1c) and the green trace isthe final result after removing the broad transparency back-ground and it is magnified 3 times for visibility purpose. / (F = 2) transitions respectively. The correspond-ing extra peaks location will be at 224 MHz and 342MHz from the main peaks. In Fig. 7b, the counter-propagating blue laser beams will form extra peaks onboth the left and right side of the main peaks. Ideallythe extra peak on the right side of the green spectrumshould vanish, but it is still visible due to incompletesubtraction. B. Resolving P / hyperfine levels in Rb
1. V-type open system
The 6P / hyperfine levels of Rb were also resolvedusing similar configurations shown in Fig. 1 and 2. Theresults of VIPO at IR plus VSS at blue transition con-figuration both in the case of a V-type open system andoptical pumping system are reported here. In this con-figuration, the probe and the counter-propagating pumplasers at 780 nm are locked to resonance on 5S / (F =2) ↔ / (F = 3) transition. The 420 nm pump laserscans across the 6P / hyperfine levels on 5S / (F = 2) ↔ / weak transition in the case of a V-type system andon 5S / (F = 1) ↔ / weak transition in the case ofoptical pumping system.The transparency spectrum of the configuration givenin Fig. 1a for Rb, is shown by the red dashed traceof Fig. 8 when all the three shutters in the experi-mental set-up of Fig. 3 are closed. The peaks of the6P / (F = 2 ,
3) hyperfine levels are well resolved butthe peaks of 6P / (F = 1 ,
2) are partially resolved dueto the residual Doppler broadening effect. When shut-ters 1 and 2 are open, the dips induced by VIPO atIR and VSS at blue transition inside the broad trans-parency peaks corresponds to the three hyperfine levels of the 6P / (F = 1 , ,
3) state (see the blue trace markedwith dots in Fig. 8). The residual Doppler broaden-ing effect is removed when shutter 3 is open and thespectrum of the resolved hyperfine levels is shown bythe green trace of Fig. 8. The linewidth of the re-solved peaks are as follows: F = 3 is 14.4 MHz, F = 2is 15.7 MHz and F = 1 is 15.8 MHz. The power ofthe pump beams labeled c1, c2 and c3 used for opti-mal signal-to-noise ratio of the spectrum are 302 . µ W(or peak intensity I=12.8 mW / cm ), 4.82 mW (or peakintensity I=102.3 mW / cm ) and 13.2 mW (or peak in-tensity I=280.1 mW / cm ) respectively.
2. Optical pumping system
The EA spectrum of the optical pumping system isshown by the red dashed trace in Fig. 9 when all thethree shutters in the experimental set-up of Fig. 3are closed. The absorption peaks corresponds to the6P / (F = 0 , ,
2) hyperfine levels in Rb. The peaksfor 6P / (F = 0 ,
1) are completely merged while thepeaks for 6P / (F = 1 ,
2) are partially merged. The lev-els 6P / (F = 1 ,
2) are detected by the probe via both thedirect decay and indirect decay channels [48] while level6P / (F = 0) is detected via the indirect decay chan-nels to 5S / (F = 2) only. When shutters 1 and 2 areopen, dips corresponding to the hyperfine levels are in-duced inside the broad EA peaks due to VIPO at IRand VSS at blue transition (see the blue trace markedwith dots in Fig. 9). The dips appear small due tobroad EA background caused by the residual Dopplerbroadening effect. The broad EA background is removedwhen shutter 3 is open and the dips corresponding tothe hyperfine levels 6P / (F = 0 , ,
2) are resolved (seethe green trace of Fig. 9). The linewidth of the re-solved peaks are as follows: F = 2 is 16 . . . . µ W(or peak intensity I=35.0 mW / cm ), 4.82 mW (or peakintensity I=102.3 mW / cm ) and 15.3 mW (or peak in-tensity I=325.5 mW / cm ) respectively.
3. Power broadening effect
The contribution of the IR pump power broadeningeffect to the final result (i.e. the resolved spectrum of the6P / state), is illustrated in Fig. 10a. The configurationused here is given in Fig. 2b (i.e. VIPO at IR transition)for the case of Rb. The power of the blue pump laserbeams is fixed (i.e. c2 is 4.26 mW and C3 is 3.27 mW)as the the power of IR pump is changed. At 1 mW ofthe IR pump, all the three peaks corresponding to the6P / (F = 0 , ,
2) hyperfine levels are well resolved (seethe red trace of Fig. 10a). However, as the IR pumppower is increased to 5 mW, the peaks corresponding0 -500 -400 -300 -200 -100 0 100 200 300 400 500
Frequency (MHz) of 420 nm laser P r ob e a b s o r p t i on ( a r b . un i t s ) Fig. 2aFig. 2cFig. 2c - Fig. 2a
Figure 9: (Color online). The EA spectrum of the 6P / hyperfine levels in Rb recorded for similar configurations of Rb shown in Fig. 2. The red dashed trace is for the opticalpumping system (Fig. 2a), the blue trace marked with dotsis for the optical pumping system with VIPO effect at IRand VSS effect at blue transition (Fig. 2c) and the greentrace is the final result after removing the broad absorptionbackground and it is magnified 3 times for visibility purpose to the 6P / (F = 0 ,
1) are completely merged as shownby the cyan trace marked by circles in Fig. 10a. Highintensity of the IR pump broadens the VIPO dips andlimits the resolution of the closely spaced hyperfine levelsof F = 0 and F = 1 which are 23.739 MHz apart [14]. Thefrequency scaling of the spectra in Fig. 10a is assignedusing the resolved peak locations of F = 1 and F = 2of the red trace. The variation of the linewidth of theresolved peak corresponding to F = 2 with the IR pumppower is shown in Fig. 10b.
V. CONCLUSIONS
In conclusion we have presented a detailed experi-mental technique to eliminate the residual (or partial)Doppler broadening in a Doppler mismatched double res-onance spectroscopy for a transparency spectrum (or en-hanced absorption spectrum). The residual two-photonDoppler broadening is removed using the VIPO at IRtransition, VSS at blue transition and the combinationof the two effects followed by the subtraction of the broadtransparency background or EA background. The tech-nique has been used to resolve the closely spaced hyper-fine levels of weak transitions for a Doppler mismatcheddouble resonance at 780 nm and 420 nm in Rb at roomtemperature.
Acknowledgement
E.O.N. would like to acknowledge Indian Council forCultural Relations (ICCR) for the PhD scholarship. K.P.would like to acknowledge the funding from SERB ofgrant No. ECR/2017/000781. -500 -400 -300 -200 -100 0 100 200 300 400 500
Frequency (MHz) of 420 nm laser P r ob e a b s o r p t i on ( a r b . un i t s ) Power (mW) of 780 nm pump laser L i n e w i d t h o f F = , R b ( M H z ) (b) Figure 10: (Color online). (a) The result of the final spectra ofthe VIPO dips of the 6P / hyperfine levels in Rb recordedfor various powers of IR pump laser after removing the broadabsorption background. (b) The variation of the linewidth ofthe resolved peak F = 2 with the power of IR pump.
Appendices
Appendix A: Transparency for V-type open system1. V-type open system
The Hamiltonian H of a V-type open system shown inFig. 1a under electric-dipole and rotating-wave approxi-mation and in the interaction picture is given as follows,H = (cid:126) (cid:8) Ω p | (cid:105)(cid:104) | + Ω c | (cid:105)(cid:104) | − ∆ p | (cid:105)(cid:104) | − ∆ c | (cid:105)(cid:104) | + h.c. (cid:9) (A1)where the levels are 5S / (F = 3) = | (cid:105) , 5P / (F = 4) = | (cid:105) , 6P / (F = 2) = | (cid:105) and 5S / (F = 2) = | (cid:105) . Theequations of motion of the density matrix are obtained1using Eq. 2 and A1 and are given as follows,˙ ρ = i Ω p ρ − ρ ) − i Ω c ρ − γ ρ ˙ ρ = i Ω c ρ − ρ ) − γ ρ − i Ω p ρ ˙ ρ = − i Ω c ρ − γ ρ − i Ω p ρ ˙ ρ = i Ω p ρ − i Ω ∗ p ρ − Γ ρ ˙ ρ = − i Ω ∗ p ρ + i Ω c ρ − γ ρ ˙ ρ = − i Ω ∗ p ρ − γ ρ ˙ ρ = − i Ω ∗ c ρ + i Ω c ρ − Γ ρ (A2)˙ ρ = − i Ω ∗ c ρ − γ ρ ˙ ρ =Γ ρ + Π g ( ρ − ρ ) where, γ = i ∆ p + γ dec , γ = i ∆ c + γ dec , γ = γ dec , γ = i (∆ c − ∆ p ) + γ dec , γ = − i ∆ p + γ dec , γ = − i ∆ c + γ dec , Γ = Γ = Π g , Γ = Γ + Γ , and γ decij = (Γ i + Γ j ), Γ i is the decay rate of the i th level, Γ andΓ are the decay rates of level 3 to level 1 and level 4respectively. The remaining density matrix equations areobtained using population conservation law (cid:80) j =1 ρ jj = 1and the complex conjugate ˙ ρ ji = ˙ ρ ∗ ij . In the steady statecondition ( ˙ ρ ij = 0 for all i and j ), the imaginary part of ρ corresponds to the absorption of the probe laser andin the weak probe approximation it is given as follows, ρ = i Π g Ω p (Γ ((Γ + Π g ) + 4∆ c ) + Ω c (Γ (Γ +Π g ) − i ∆ c (2Γ +Π g )+2 i ∆ p (Γ +Π g ))Γ +Γ − i (∆ c − ∆ p ) )(Ω c (Γ + Π g )(Γ + 3Π g ) + 2Γ Π g ((Γ + Π g ) + 4∆ c ))( Ω c Γ +Γ − i ∆ c +2 i ∆ p + Γ + Π g + 2 i ∆ p ) (A3) -100 -50 0 50 100 Detuning of 420 nm laser (MHz) P r ob e a b s o r p t i on -4 NumericalAnalytical
Figure A.1: (Color online). Comparison of the full numericalsolution of the density matrix in a V-type open system withthe the analytical solution given in Eq. A3. Ω c = √ . ,Γ = 2 π × .
065 MHz , Γ = 2 π × .
32 MHz.
The solution of a V-type open system given in Eq. A3 isgraphically represented in Fig. A.1 and is well matchedwith the numerical simulation of the full density matrixgiven in Eq. A2.
2. VIPO at IR transition for V-type open system
The Hamiltonian of the VIPO at IR transition for aV-type open system shown in Fig. 1b is given in Eq. 1. The following set of equations of motion are obtained bysubstitution of Eq. 1 into Eq. 2. ˙ ρ = i c1 + Ω p e iδ t )( ρ − ρ ) − i Ω c2 ρ − γ ρ ˙ ρ = i Ω c2 ρ − ρ ) − γ ρ − i c1 + Ω p e iδ t ) ρ ˙ ρ = − i Ω c2 ρ − γ ρ − i c1 + Ω p e iδ t ) ρ ˙ ρ = i c1 + Ω p e iδ t ) ρ − i ∗ c1 + Ω ∗ p e − iδ t ) ρ − Γ ρ ˙ ρ = − i ∗ c1 + Ω ∗ p e − iδ t ) ρ + i Ω c2 ρ − γ ρ ˙ ρ = − i ∗ c1 + Ω ∗ p e − iδ t ) ρ − γ ρ ˙ ρ = − i Ω ∗ c2 ρ + i Ω c2 ρ − Γ ρ (A4)˙ ρ = − i Ω ∗ c2 ρ − γ ρ ˙ ρ =Γ ρ + Π g ( ρ − ρ ) where, γ = i ∆ c1 + γ dec , γ = i ∆ c2 + γ dec , γ = γ dec , γ = i (∆ c2 − ∆ c1 ) + γ dec , γ = − i ∆ c1 + γ dec , γ = − i ∆ c2 + γ dec . The steady state solution of theequations of motion given in Eq. A4 is given in Eq. 4and the individual contribution of the terms I, II and IIIis illustrated in Fig. A.2.2 -100 -50 0 50 100 Detuning of 420 nm laser (MHz) P r ob e bb s o r p t i on -4 P r ob e bb s o r p t i on -5 I+II+IIIIIIIII
Figure A.2: (Color online). The graphical representation ofthe individual terms I, II and III given in Eq. 4. Ω c = Γ ,Ω c = √ . , Γ = 2 π × .
065 MHz , Γ = 2 π × .
32 MHz.The vertical axis of the red trace marked by circles and theblue dashed trace are on the left and the green trace markedwith dots and the cyan trace are on the right.
3. VIPO at IR and VSS at blue transition forV-type open system
The Hamiltonian of the VIPO at IR transition for a V-type open system considered in Fig. 1b is given in Eq. 6.The following set of equations of motion of the densitymatrix are similarly obtained by substitution of Eq. 6into Eq. 2. ˙ ρ = i c1 + Ω p e iδ t )( ρ − ρ ) − γ ρ − i Ω c2 e iδ t ) ρ (A5)˙ ρ = i Ω c2 e iδ t )( ρ − ρ ) − γ ρ − i c1 + Ω p e iδ t ) ρ ˙ ρ = − i Ω c2 e iδ t ) ρ − γ ρ − i c1 + Ω p e iδ t ) ρ ˙ ρ = i c1 + Ω p e iδ t ) ρ − i ∗ c1 + Ω ∗ p e − iδ t ) ρ − Γ ρ ˙ ρ = − i ∗ c1 + Ω ∗ p e − iδ t ) ρ + i Ω c2 e iδ t ) ρ − γ ρ ˙ ρ = − i ∗ c1 + Ω ∗ p e − iδ t ) ρ − γ ρ ˙ ρ = − i Ω ∗ c2 e − iδ t ) ρ + i Ω c2 e iδ t ) ρ − Γ ρ ˙ ρ = − i Ω ∗ c2 e − iδ t ) ρ − γ ρ ˙ ρ =Γ ρ + Π g ( ρ − ρ ) Appendix B: Enhanced absorption for opticalpumping system1. Optical pumping system
The Hamiltonian H of the optical pumping system con-sider in Fig. 2a under electric-dipole and rotating-waveapproximation and in the interaction picture is given asfollows,H = (cid:126) (cid:8) Ω p | (cid:105)(cid:104) | + Ω c | (cid:105)(cid:104) | − ∆ p | (cid:105)(cid:104) | − ∆ c | (cid:105)(cid:104) | + h.c. (cid:9) (B1)The equations of motion of the density matrix is obtainedfrom Eq. 2 and B1 and set of equations are given asfollows,˙ ρ = i Ω p ρ − ρ ) − γ dec ρ (B2)˙ ρ = − i Ω p ρ + i Ω ∗ c ρ − γ ρ ˙ ρ = − i Ω p ρ + i Ω c ρ − γ ρ ˙ ρ = − i Ω ∗ p ρ + i Ω p ρ − Γ ρ ˙ ρ = − i Ω ∗ p ρ − γ ρ + i Ω ∗ c ρ ˙ ρ = − i Ω ∗ p ρ − γ ρ + i Ω c ρ ˙ ρ = − i Ω c ρ + i Ω ∗ c ρ − Γ ρ ˙ ρ = − i Ω c ρ − ρ ) − γ ρ ˙ ρ = − i Ω ∗ c ρ + i Ω c ρ + Γ ρ + Π g ( ρ − ρ )where, γ = i ∆ p + γ dec , γ = γ dec , γ = i ∆ c + γ dec , γ = − i ∆ p + γ dec , γ = i (∆ c − ∆ p )+ γ dec , γ = i ∆ c + γ dec . The steady state solution of Eq. B2 in the weakprobe approximation which gives enhanced absorptionspectrum of the probe is expressed as follows, ρ = i Ω p (Ω c (Γ + Π g )(Γ + Π g ) + Γ Π g ((Γ + Π g ) + 4∆ c ))(Γ + Π g + 2 i ∆ p )(Ω c (Γ + Π g )(Γ + 3Π g ) + 2Γ Π g ((Γ + Π g ) + 4∆ c )) (B3)The solution of optical pumping system given in Eq. B3is graphically represented in Fig. B.1 and is well matched with the numerical simulation of the full density matrixgiven in Eq. B2.3 -100 -50 0 50 100 Detuning of 420 nm laser (MHz) P r ob e a b s o r p t i on -4 NumericalAnalytical
Figure B.1: (Color online). Comparison of the full numericalsolution of the density matrix in optical pumping system withthe the analytical solution given in Eq. B3. Ω c = √ . ,Γ = 2 π × .
065 MHz , Γ = 2 π × .
32 MHz.
2. VIPO at IR transition for optical pumpingsystem
The Hamiltonian of the VIPO at IR transition for theoptical pumping system consider in Fig. 2b is given in Eq.9. The equations of motion of density matrix elements isalso obtained from Eq. 2 and 9 which gives the followingset of equations. ˙ ρ = i c + Ω p e iδ t )( ρ − ρ ) − γ ρ (B4)˙ ρ = − i c + Ω p e iδ t ) ρ + i Ω ∗ c ρ − γ ρ ˙ ρ = − i c + Ω p e iδ t ) ρ + i Ω c ρ − γ ρ ˙ ρ = − i ∗ c + Ω ∗ p e − iδ t ) ρ + i c + Ω p e iδ t ) ρ − Γ ρ ˙ ρ = − i ∗ c + Ω ∗ p e − iδ t ) ρ − γ ρ + i Ω ∗ c ρ ˙ ρ = − i ∗ c + Ω ∗ p e − iδ t ) ρ − γ ρ + i Ω c ρ ˙ ρ = − i Ω c ρ + i Ω ∗ c ρ − Γ ρ ˙ ρ = − i Ω c ρ − ρ ) − γ ρ ˙ ρ = − i Ω ∗ c ρ + i Ω c ρ + Γ ρ + Π g ( ρ − ρ ) The steady state solution of the equations of motiongiven in Eq. B4 is given in Eq. 10 and the solution of thevarious density matrix components ρ (0)11 , ρ (0)22 , ρ (+1)11 and ρ (+1)22 is given as follows. The individual contribution ofthe terms I and II given in Eq. 10 is also illustrated inFig. B.2.
3. VIPO at IR and VSS at blue transition foroptical pumping system
The Hamiltonian of the VIPO at IR transition andVSS at blue transition for the optical pumping systemconsider in Fig. 2c is given in Eq. 11. The equations of -100 -50 0 50 100
Detuning of 420 nm laser (MHz) P r ob e a b s o r p t i on -4 P r ob e a b s o r p t i on -4 I+IIIII
Figure B.2: (Color online). The graphical representation ofthe individual terms I and II given in Eq. 10. Ω c = Γ ,Ω c = √ . , Γ = 2 π × .
065 MHz , Γ = 2 π × .
32 MHz.The vertical axis of the red trace marked by circles is on theleft and the blue dashed trace and the green trace markedwith dots are on the right. motion of the density matrix is obtained from Eq. 2 and11 as follows. ˙ ρ = i c + Ω p e iδ t )( ρ − ρ ) − γ dec ρ (B5)˙ ρ = − i c + Ω p e iδ t ) ρ + i Ω ∗ c e − iδ t ) ρ − γ ρ ˙ ρ = − i c + Ω p e iδ t ) ρ + i Ω c e iδ t ) ρ − γ ρ ˙ ρ = − i ∗ c + Ω ∗ p e − iδ t ) ρ + i c + Ω p e iδ t ) ρ − Γ ρ ˙ ρ = − i ∗ c + Ω ∗ p e − iδ t ) ρ + i Ω ∗ c e − iδ t ) ρ − γ ρ ˙ ρ = − i ∗ c + Ω ∗ p e − iδ t ) ρ + i Ω c e iδ t ) ρ − γ ρ ˙ ρ = − i Ω c e iδ t ) ρ + i Ω ∗ c e − iδ t ) ρ − Γ ρ ˙ ρ = − i Ω c e iδ t )( ρ − ρ ) − γ ρ ˙ ρ = − i Ω ∗ c e − iδ t ) ρ + i Ω c e iδ t ) ρ + Γ ρ + Π g ( ρ − ρ )[1] K.-B. Im, H.-Y. Jung, C.-H. Oh, S.-H. Song, P.-S.Kim, and H.-S. Lee, Phys. Rev. A , 034501 (2001),URL https://link.aps.org/doi/10.1103/PhysRevA.63.034501 . [2] K.-J. Boller, A. Imamo˘glu, and S. E. Harris, Phys. 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